Theorem bnj156 31024

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj156.1	⊢ (𝜁0 ↔ (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′))
bnj156.2	⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
bnj156.3	⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
bnj156.4	⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)

Assertion

Ref	Expression
bnj156	⊢ (𝜁1 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))

Proof of Theorem bnj156

Step	Hyp	Ref	Expression
1		bnj156.2	. 2 ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
2		bnj156.1	. . . 4 ⊢ (𝜁0 ↔ (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′))
3	2	sbcbii 3597	. . 3 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′))
4		sbc3an 3600	. . . 4 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1𝑜 ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′))
5		bnj62 31016	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 1𝑜 ↔ 𝑔 Fn 1𝑜)
6		bnj156.3	. . . . . 6 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
7	6	bicomi 214	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ 𝜑1)
8		bnj156.4	. . . . . 6 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
9	8	bicomi 214	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ 𝜓1)
10	5, 7, 9	3anbi123i 1388	. . . 4 ⊢ (([𝑔 / 𝑓]𝑓 Fn 1𝑜 ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))
11	4, 10	bitri 264	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))
12	3, 11	bitri 264	. 2 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))
13	1, 12	bitri 264	1 ⊢ (𝜁1 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))

Syntax hints:   ↔ wb 196   ∧ w3a 1072  [wsbc 3541   Fn wfn 5996  1_𝑜c1o 7673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-fun 6003  df-fn 6004

This theorem is referenced by:  bnj153  31178