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Theorem bj-gabeqd 36938
Description: Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
Hypotheses
Ref Expression
bj-gabeqd.nf (𝜑 → ∀𝑥𝜑)
bj-gabeqd.c (𝜑𝐴 = 𝐵)
bj-gabeqd.f (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
bj-gabeqd (𝜑 → {𝐴𝑥𝜓} = {𝐵𝑥𝜒})

Proof of Theorem bj-gabeqd
StepHypRef Expression
1 bj-gabeqd.nf . . 3 (𝜑 → ∀𝑥𝜑)
2 bj-gabeqd.c . . 3 (𝜑𝐴 = 𝐵)
3 bj-gabeqd.f . . . 4 (𝜑 → (𝜓𝜒))
43biimpd 229 . . 3 (𝜑 → (𝜓𝜒))
51, 2, 4bj-gabssd 36937 . 2 (𝜑 → {𝐴𝑥𝜓} ⊆ {𝐵𝑥𝜒})
62eqcomd 2743 . . 3 (𝜑𝐵 = 𝐴)
73biimprd 248 . . 3 (𝜑 → (𝜒𝜓))
81, 6, 7bj-gabssd 36937 . 2 (𝜑 → {𝐵𝑥𝜒} ⊆ {𝐴𝑥𝜓})
95, 8eqssd 4001 1 (𝜑 → {𝐴𝑥𝜓} = {𝐵𝑥𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  {bj-cgab 36934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ss 3968  df-bj-gab 36935
This theorem is referenced by: (None)
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