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Theorem bnj1536 32234
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1536.1 (𝜑𝐹 Fn 𝐴)
bnj1536.2 (𝜑𝐺 Fn 𝐴)
bnj1536.3 (𝜑𝐵𝐴)
bnj1536.4 (𝜑 → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
Assertion
Ref Expression
bnj1536 (𝜑 → (𝐹𝐵) = (𝐺𝐵))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem bnj1536
StepHypRef Expression
1 bnj1536.4 . 2 (𝜑 → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
2 bnj1536.1 . . 3 (𝜑𝐹 Fn 𝐴)
3 bnj1536.2 . . 3 (𝜑𝐺 Fn 𝐴)
4 bnj1536.3 . . 3 (𝜑𝐵𝐴)
5 fvreseq 6791 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
62, 3, 4, 5syl21anc 836 . 2 (𝜑 → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
71, 6mpbird 260 1 (𝜑 → (𝐹𝐵) = (𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wral 3109  wss 3884  cres 5525   Fn wfn 6323  cfv 6328
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  bnj1523  32451
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