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Theorem fveq1 5327

Description: Equality theorem for function value. (Contributed by set.mm contributors, 29-Dec-1996.)

**Assertion**
Ref	Expression
fveq1	⊢ (F = G → (F ‘A) = (G ‘A))

Proof of Theorem fveq1 Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 4641	. . 3 ⊢ (F = G → (AFx ↔ AGx))
2	1	iotabidv 4360	. 2 ⊢ (F = G → (℩xAFx) = (℩xAGx))
3		df-fv 4795	. 2 ⊢ (F ‘A) = (℩xAFx)
4		df-fv 4795	. 2 ⊢ (G ‘A) = (℩xAGx)
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (F = G → (F ‘A) = (G ‘A))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1642 ℩cio 4337 class class class wbr 4639 ‘cfv 4781

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-uni 3892 df-iota 4339 df-br 4640 df-fv 4795

This theorem is referenced by: fveq1i 5329 fveq1d 5330 eqfnfv 5392 isoeq1 5482 oveq 5529