Theorem bj-cbvalvv 33063

Description: Version of cbvalv 2433 with a dv condition, which does not require ax-13 2407. UPDATE: this is cbvalvw 2124 (which is proved with fewer axioms). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-cbvalvv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-cbvalvv	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem bj-cbvalvv

Step	Hyp	Ref	Expression
1		nfv 1994	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1994	. 2 ⊢ Ⅎ𝑥𝜓
3		bj-cbvalvv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvalv1 2335	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857

This theorem is referenced by:  bj-zfpow  33125  bj-nfcjust  33176