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Mirrors > Home > NFE Home > Th. List > rexeq | GIF version |
Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) |
Ref | Expression |
---|---|
rexeq | ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2489 | . 2 ⊢ ℲxA | |
2 | nfcv 2489 | . 2 ⊢ ℲxB | |
3 | 1, 2 | rexeqf 2804 | 1 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∃wrex 2615 |
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-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 |
This theorem is referenced by: rexeqi 2812 rexeqdv 2814 rexeqbi1dv 2816 unieq 3900 xpkeq1 4198 xpkeq2 4199 imakeq2 4225 tfineq 4488 nnadjoin 4520 tfinnn 4534 imaeq2 4938 qseq1 5974 brlecg 6112 ovmuc 6130 tceq 6158 lec0cg 6198 sbth 6206 dflec3 6221 lenc 6223 |
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