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Theorem bnj956 31175
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj956.1 (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
Assertion
Ref Expression
bnj956 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Proof of Theorem bnj956
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bnj956.1 . . . 4 (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
2 eleq2 2828 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
32anbi1d 743 . . . . . 6 (𝐴 = 𝐵 → ((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)))
43alexbii 1909 . . . . 5 (∀𝑥 𝐴 = 𝐵 → (∃𝑥(𝑥𝐴𝑦𝐶) ↔ ∃𝑥(𝑥𝐵𝑦𝐶)))
5 df-rex 3056 . . . . 5 (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥(𝑥𝐴𝑦𝐶))
6 df-rex 3056 . . . . 5 (∃𝑥𝐵 𝑦𝐶 ↔ ∃𝑥(𝑥𝐵𝑦𝐶))
74, 5, 63bitr4g 303 . . . 4 (∀𝑥 𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐶))
81, 7syl 17 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐶))
98abbidv 2879 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐶})
10 df-iun 4674 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
11 df-iun 4674 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐶}
129, 10, 113eqtr4g 2819 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wrex 3051   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rex 3056  df-iun 4674
This theorem is referenced by:  bnj1316  31219  bnj953  31337  bnj1000  31339  bnj966  31342
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