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Mirrors > Home > MPE Home > Th. List > ceqex | Structured version Visualization version GIF version |
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.) |
Ref | Expression |
---|---|
ceqex | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2144 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
2 | 1 | ex 413 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
3 | eqvisset 3454 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) | |
4 | alexeqg 3583 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
6 | sp 2146 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜑)) | |
7 | 6 | com12 32 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜑)) |
8 | 5, 7 | sylbird 261 | . 2 ⊢ (𝑥 = 𝐴 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜑)) |
9 | 2, 8 | impbid 213 | 1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1520 = wceq 1522 ∃wex 1761 ∈ wcel 2081 Vcvv 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-v 3439 |
This theorem is referenced by: ceqsexg 3585 |
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