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Theorem bnj956 32055
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 2900 . . . . . . 7 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
32anbi1d 632 . . . . . 6 (𝐴 = 𝐵 → ((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)))
43alexbii 1834 . . . . 5 (∀𝑥 𝐴 = 𝐵 → (∃𝑥(𝑥𝐴𝑦𝐶) ↔ ∃𝑥(𝑥𝐵𝑦𝐶)))
5 df-rex 3132 . . . . 5 (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥(𝑥𝐴𝑦𝐶))
6 df-rex 3132 . . . . 5 (∃𝑥𝐵 𝑦𝐶 ↔ ∃𝑥(𝑥𝐵𝑦𝐶))
74, 5, 63bitr4g 317 . . . 4 (∀𝑥 𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐶))
81, 7syl 17 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝐶 ↔ ∃𝑥𝐵 𝑦𝐶))
98abbidv 2885 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐶})
10 df-iun 4894 . 2 𝑥𝐴 𝐶 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐶}
11 df-iun 4894 . 2 𝑥𝐵 𝐶 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝐶}
129, 10, 113eqtr4g 2881 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  {cab 2799  wrex 3127   ciun 4892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-rex 3132  df-iun 4894
This theorem is referenced by:  bnj1316  32099  bnj953  32218  bnj1000  32220  bnj966  32223
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