Theorem frege55lem2c 43879

Description: Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55lem2c	⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem frege55lem2c

Step	Hyp	Ref	Expression
1		vex 3492	. . 3 ⊢ 𝑥 ∈ V
2	1	frege54cor1c 43877	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
3		frege53c 43876	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥))
4	2, 3	ax-mp 5	1 ⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3488  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-frege8 43771  ax-frege52c 43850

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-sbc 3805  df-sn 4649

This theorem is referenced by: (None)