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Theorem bnj956 29903
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 2672 . . . . . . . 8 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
32anbi1d 736 . . . . . . 7 (𝐴 = 𝐵 → ((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)))
43alimi 1728 . . . . . 6 (∀𝑥 𝐴 = 𝐵 → ∀𝑥((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)))
5 exbi 1760 . . . . . 6 (∀𝑥((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)) → (∃𝑥(𝑥𝐴𝑦𝐶) ↔ ∃𝑥(𝑥𝐵𝑦𝐶)))
64, 5syl 17 . . . . 5 (∀𝑥 𝐴 = 𝐵 → (∃𝑥(𝑥𝐴𝑦𝐶) ↔ ∃𝑥(𝑥𝐵𝑦𝐶)))
7 df-rex 2897 . . . . 5 (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥(𝑥𝐴𝑦𝐶))
8 df-rex 2897 . . . . 5 (∃𝑥𝐵 𝑦𝐶 ↔ ∃𝑥(𝑥𝐵𝑦𝐶))
96, 7, 83bitr4g 301 . . . 4 (∀𝑥 𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐶))
101, 9syl 17 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐶))
1110abbidv 2723 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐶})
12 df-iun 4447 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
13 df-iun 4447 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐶}
1411, 12, 133eqtr4g 2664 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1975  {cab 2591  wrex 2892   ciun 4445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-rex 2897  df-iun 4447
This theorem is referenced by:  bnj1316  29947  bnj953  30065  bnj1000  30067  bnj966  30070
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