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| Mirrors > Home > MPE Home > Th. List > rexeqbi1dv | Structured version Visualization version GIF version | ||
| Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) (Proof shortened by Steven Nguyen, 5-May-2023.) |
| Ref | Expression |
|---|---|
| raleqbi1dv.1 | ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| rexeqbi1dv | ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
| 2 | raleqbi1dv.1 | . 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
| 3 | 1, 2 | rexeqbidvv 3336 | 1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wrex 3070 |
| 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-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-rex 3071 |
| This theorem is referenced by: rexeqOLD 3341 friOLD 5643 frsn 5773 isofrlem 7360 f1oweALT 7997 frxp 8151 frxp2 8169 1sdomOLD 9285 oieq2 9553 zfregcl 9634 frmin 9789 hashge2el2difr 14520 cat1 18142 ishaus 23330 isreg 23340 isnrm 23343 lebnumlem3 24995 1vwmgr 30295 3vfriswmgr 30297 isgrpo 30516 pjhth 31412 bnj1154 35013 satfvsuc 35366 satf0suc 35381 sat1el2xp 35384 fmlasuc0 35389 isexid2 37862 ismndo2 37881 rngomndo 37942 relpfrlem 44974 stoweidlem28 46043 prprval 47501 |
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