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Mirrors > Home > MPE Home > Th. List > rexeqOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rexeq 3405 as of 5-May-2023. Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rexeqOLD | ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2976 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2976 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rexeqf 3397 | 1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∃wrex 3138 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rex 3143 |
This theorem is referenced by: (None) |
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